A new proof of Sárközy's theorem

Author:
Neil Lyall

Journal:
Proc. Amer. Math. Soc. **141** (2013), 2253-2264

MSC (2010):
Primary 11B30

DOI:
https://doi.org/10.1090/S0002-9939-2013-11628-X

Published electronically:
March 14, 2013

MathSciNet review:
3043007

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Abstract: It is a striking and elegant fact (proved independently by Furstenberg and Sárközy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect square. In this article we present a new and simple proof of this result by adapting an argument originally developed by Croot and Sisask to give a new proof of Roth's theorem.

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Additional Information

**Neil Lyall**

Affiliation:
Department of Mathematics, The University of Georgia, Athens, Georgia 30602

Email:
lyall@math.uga.edu

DOI:
https://doi.org/10.1090/S0002-9939-2013-11628-X

Received by editor(s):
October 18, 2011

Published electronically:
March 14, 2013

Dedicated:
Dedicated to Steve Wainger on the occasion of his retirement

Communicated by:
Michael T. Lacey

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.